Let Atf(x) =

f(x − ty)dσ(y), where S is a smooth compact hypersurface in Rn and dσ denotes the Lebesgue measure on S. Let Af(x) = supt>0 |Atf(x)|. If the hypersurface S has non-vanishing Gaussian curvature, then (*) ||Af ||Lp(Rn) ≤ Cp||f ||Lp(Rn), f ∈ S(Rn), for p> nn−1. Moreover, the result is sharp. See [St76], [Gr82]. If the hypersurface S is convex and the order of contact with every tangent line is finite, the optimal exponents for the inequality (∗) are known in R3, (see [IoSaSe97]), and in any dimension in the range p> 2, (see [IoSa96]). More precisely, the result in the range p> 2 is the following. Theorem 1 ([IoSa96]). Let S be a smooth convex compact finite type hypersurface, in the sense that the order of contact with every tangent line is finite. Then for p> 2, the following condition is necessary and sufficient for the maximal inequality (∗). (1) (d(x,H))−1 ∈ L 1p (S), for every tangent hyperplane H not passing through the origin, where d(x,H) denotes the distance from a point x ∈ S to the tangent hyperplane H. In fact, the condition (1) is a necessary condition for any smooth compact hypersurface in Rn. See [IoSa96], Theorem 2. In this paper we shall consider convex radial hypersurfaces of the form (2) S = {x ∈ B: xn = γ(|x′|) + 1}, where B is a ball centered at the origin, x = (x′, xn), γ is convex, γ, γ′ ′ increasing, γ(0) = γ′(0) = 0, and γ is allowed to vanish of infinite order